Alphabeta MathHopf Ideals, Finite Duals, and Basic Constructions — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Categorification and Geometric Representation Theory
Commutative Algebra
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Example
The dual Hopf algebra of a finite group algebra
Example
Coideals, ring ideals, Hopf ideals, and subbialgebras
Example
Sweedler's four-dimensional Hopf algebra
Example
The finite dual of the primitive polynomial Hopf algebra
Example
A tensor product Hopf algebra from a group and a primitive element
Sources
- Andrew Baker, Hopf algebras over fields: an introduction, §3.4
- Andrew Baker, Hopf algebras over fields: an introduction, §§2.23–2.24, §3.1, and §3.4
- Andrew Baker, Hopf algebras over fields: an introduction, §3.12
- Kalle Kytölä, Introduction to Hopf algebras and representations, §§3.8–3.9
- Kalle Kytölä, Introduction to Hopf algebras and representations, Exercise 28